Optimized aerodynamic profile for a turbine blade, in particular for a rotary wheel of the fourth stage of a turbine

ABSTRACT

When cold and in the non-coated state, the aerodynamic profile is substantially identical to a nominal profile determined by the Cartesian coordinates X,Y, Zadim given in Table 1, in which the coordinate Zadim is the quotient D/H where D is the distance of the point under consideration from a first reference plane P0 situated at the base of the nominal profile, and H is the height of said profile measured from the first reference plane to a second reference plane P1. The measurements D and H are taken radially relative to the axis of the turbine, while the X coordinate is measured in the axial direction of the turbine.

CROSS-REFERENCE TO RELATED APPLICATION(S)

This patent application claims the benefit under 35 U.S.C. §119 to U.S.Provisional Application No. 62/361,539, filed on Jul. 13, 2016, theentirety of which is incorporated herein by reference.

FIELD OF THE DISCLOSURE

The present invention relates to an aerodynamic profile for a turbineblade.

INTRODUCTION

In particular, the invention relates to the aerodynamic profile of ablade of a rotary wheel of a gas turbine, and more particularly of a lowpressure turbine of the type used in a turbojet of an aircraft.

The invention relates in particular to a blade of the rotary wheel ofthe fourth stage of a turbine having a plurality of stages, preferablyseven stages.

Such a profile should enable the turbine to provide the desiredefficiency, and in order to do that it must be such that the flow of airaround the profile is sound, i.e. substantially such that it does notgive rise to turbulence, which is harmful for overall efficiency.

It must be capable of being installed properly in the environment of theengine, and in particular, for the blade of a rotary wheel forming aportion of the rotor of the turbine, it must be capable of beingfastened easily to the support disk of the rotary wheel, via the root ofthe blade, whereas its tip must be capable of carrying a top platform.Furthermore, this part must present a profile enabling it to bemanufactured in a reliable and cost-effective manner by availablemanufacturing methods, such as casting, forging, machining, additivefabrication, or else weaving, without this list being limitative.

Furthermore, the profile of the blade must enable it to withstand themechanical stresses to which the blade is subjected, by enabling thosestresses to be spread over the entire blade in such a manner as to avoidpremature wear thereof. This spreading must apply both in static modeand in dynamic mode. In particular, it must be such that the blade,given its resonant modes, can withstand the highest dynamic loading forgood operation of the turbine.

An object of the invention is to propose an aerodynamic profile for aturbine blade that is optimized, and capable of satisfying thoseobjectives.

This object is achieved by the fact that when cold and in a non-coatedstate, said profile is substantially identical to a nominal profiledetermined by the Cartesian coordinates X,Y,Zadim given in Table 1, inwhich the coordinate Zadim is the quotient D/H where D is the distanceof the point P under consideration from a reference X,Y plane situatedat the base of the nominal profile, and H is the height of said profilemeasured from said reference plane that is the intersection of thestacking axis of the set of blades and the axisymmetric surface of thehub, out to a second reference plane that is the intersection of saidstacking axis with the axisymmetric surface of the casing, themeasurements D and H being taken radially relative to the axis of theturbine, while the coordinate X is measured in the axial direction ofthe turbine.

This profile has been determined as a result of numerous tests andsimulations. It is defined cold, i.e. at an ambient temperature of 20°C. That is a reference temperature at which the profile is geometricallydetermined. The above-mentioned objectives of aerodynamics andmechanical optimization are naturally valid for the conditions of use ofthis aerodynamic profile, i.e. when hot, at a temperature that isstabilized when the engine of which the turbine forms a part is beingused under cruising conditions.

Furthermore, the aerodynamic profile of the invention is defined in thenon-coated state. Since turbine blades are subjected to high temperaturegradients, it is common practice for them to be provided with a coatinghaving thermal properties enabling them more easily to withstand suchtemperature variations. The profile is determined prior to installingsuch a coating.

It is stated above that the profile of the invention is “substantiallyidentical” to the nominal profile. This means that the profile maydepart very slightly from said nominal profile.

The aerodynamic profile is thus preferably defined within an envelope of±1 millimeter (mm) in a direction normal to the surface of the nominalprofile.

This variation takes account in particular of manufacturing tolerancesof the profile.

It is also preferable for the X,Y coordinates of the aerodynamic profileto lie within a range of ±5% relative to the X,Y coordinates of thenominal profile.

This variation takes account of the setting of the profile to adapt tothe flow coming from the turbine vanes situated upstream, so as tofurther improve the efficiency of the turbine. In particular, theprofile is adapted so as to be optimized for the flow coming from thenozzle of the fourth stage of a turbine, when the blade of the inventionbelongs to the rotary wheel of the fourth stage of a turbine.

For a blade of a rotary wheel, forming a portion of the rotor of aturbine, the profile serves to optimize the inter-blade flow, i.e. theflow between adjacent blades of the wheel, and thus transforms a maximumamount of the kinetic energy of the gaseous fluid into kinetic energy onthe shaft of the turbine. In addition, it serves to orient the gaseousfluid correctly for the turbine vanes situated downstream, in particularbelonging to the stationary nozzle of the next stage of the turbine, ifsuch a next stage exists.

BRIEF DESCRIPTION OF THE DRAWINGS

The invention can be better understood and its advantages appear betteron reading the following detailed description of an embodiment given byway of non-limiting example. The description refers to the accompanyingdrawings, in which:

FIG. 1 is a fragmentary axial section view of a turbine including ablade of profile that corresponds to that of the invention;

FIG. 2 is an enlarged view of FIG. 1, showing the fourth stage of theturbine.

DETAILED DESCRIPTION

The turbine shown in axial section in FIG. 1 comprises seven stagesreferenced respectively S1 to S7. In the direction DF going fromupstream to downstream, each stage comprises a nozzle that forms aportion of the stator of the turbine and that has a plurality ofradially-oriented vanes, and a rotary wheel that forms a portion of therotor of the turbine and that likewise includes a plurality ofradially-oriented blades.

In FIG. 1, the vanes of the nozzles of stages S1 to S7 are givenreferences N1 to N7, whereas the blades of rotary wheels of stages S1 toS7 are given respective references W1 to W7.

In known manner, the vanes of the nozzles are fastened at both ends tostationary structure portions, while the blades of the rotary wheels arefastened to a rotary disk via their roots that are formed at theirradially-inner ends closer to the axis of the rotation A of the turbine.The various disks designated by respective references d1 to d7 areconstrained to rotate together. In the example shown, the tips of theblades of the rotary wheels, formed at their radially-outer ends, carrytop platforms given respective references t1 to t4.

The invention relates in particular to the blade W4 which is a blade ofthe rotary wheel of the fourth stage S4 of the turbine.

The blade W4 shown in FIG. 2 includes, at its radially-inner end, ablade root 10 enabling it to be connected to a support disk forming thehub of a turbine wheel, and a platform 12 from which an aerodynamicprofile 14 extends. At its tip, formed at its radially-outer end remotefrom the platform 12, the blade includes a top platform 16 that forms anouter casing carried by the various blades of the wheel.

By convention, the aerodynamic profile 14 is the entire portion of saidblade that extends radially from the platform 12 to the top platform 16.

In FIG. 2, there can be seen a frame of reference comprising Cartesiancoordinates X,Y,Z. The radial direction Z is the height direction of theblade, which extends radially from its root to its tip. This direction Zis perpendicular to the axial direction X which is the direction of theaxis of rotation A of the turbine. The direction Y is perpendicular tothe X,Z plane and is therefore tangential to the direction of rotationof the turbine.

The nominal profile from which the aerodynamic profile of the inventionis determined is defined in following Table 1 of coordinates, in whichthe coordinates Zadim, measured along the axis Z is non-dimensional,whereas the dimensions X and Y, respectively measured along the axes Xand Y, are expressed in millimeters.

TABLE 1 X Y Zadim −9.44397 −1.47106 −0.1 −9.4865 −1.42905 −0.1 −9.53786−1.35883 −0.1 −9.59092 −1.23519 −0.1 −9.61575 −1.04908 −0.1 −9.58679−0.80499 −0.1 −9.49251 −0.51286 −0.1 −9.32959 −0.17996 −0.1 −9.095360.187504 −0.1 −8.7875 0.583416 −0.1 −8.40364 1.000727 −0.1 −7.94141.430854 −0.1 −7.39866 1.863178 −0.1 −6.77351 2.284114 −0.1 −6.064832.676393 −0.1 −5.2737 3.019011 −0.1 −4.40725 3.294721 −0.1 −3.477563.496833 −0.1 −2.49168 3.559473 −0.1 −1.4746 3.46996 −0.1 −0.455623.224363 −0.1 0.541837 2.832205 −0.1 1.491136 2.296603 −0.1 2.3805941.644151 −0.1 3.212377 0.907818 −0.1 3.987235 0.109508 −0.1 4.708858−0.73107 −0.1 5.381838 −1.59719 −0.1 6.008959 −2.47599 −0.1 6.592253−3.35682 −0.1 7.136588 −4.22817 −0.1 7.644911 −5.08093 −0.1 8.117632−5.90842 −0.1 8.557502 −6.70319 −0.1 8.966284 −7.45908 −0.1 9.344114−8.1715 −0.1 9.691337 −8.83638 −0.1 10.00977 −9.44957 −0.1 10.2984−10.0091 −0.1 10.55733 −10.5133 −0.1 10.78721 −10.9607 −0.1 10.98807−11.3515 −0.1 11.16033 −11.6861 −0.1 11.30474 −11.9661 −0.1 11.42241−12.1937 −0.1 11.51484 −12.372 −0.1 11.5583 −12.5163 −0.1 11.5306−12.6185 −0.1 11.49016 −12.6709 −0.1 11.4497 −12.7009 −0.1 11.41022−12.7186 −0.1 11.35518 −12.7295 −0.1 11.26746 −12.7172 −0.1 11.16335−12.6419 −0.1 11.08247 −12.4883 −0.1 10.97919 −12.2924 −0.1 10.85138−12.052 −0.1 10.6975 −11.7656 −0.1 10.51627 −11.4321 −0.1 10.3065−11.0516 −0.1 10.06706 −10.6248 −0.1 9.797192 −10.153 −0.1 9.496483−9.63803 −0.1 9.163177 −9.08327 −0.1 8.797617 −8.49143 −0.1 8.39865−7.86683 −0.1 7.965153 −7.21449 −0.1 7.497571 −6.53894 −0.1 6.9947−5.84649 −0.1 6.455439 −5.14415 −0.1 5.880373 −4.43821 −0.1 5.268504−3.73706 −0.1 4.604059 −3.06401 −0.1 3.878763 −2.4411 −0.1 3.103523−1.87647 −0.1 2.290575 −1.3786 −0.1 1.45415 −0.95435 −0.1 0.609836−0.60816 −0.1 −0.22715 −0.3432 −0.1 −1.05306 −0.17093 −0.1 −1.84965−0.06038 −0.1 −2.6154 −0.00692 −0.1 −3.35646 −0.00409 −0.1 −4.07743−0.05426 −0.1 −4.77265 −0.16283 −0.1 −5.43025 −0.32498 −0.1 −6.04005−0.52993 −0.1 −6.59091 −0.75973 −0.1 −7.07745 −1.01153 −0.1 −7.53−1.21911 −0.1 −7.92996 −1.40165 −0.1 −8.28089 −1.55221 −0.1 −8.59273−1.64861 −0.1 −8.86283 −1.68326 −0.1 −9.08072 −1.65451 −0.1 −9.23499−1.59323 −0.1 −9.33922 −1.54165 −0.1 −9.32465 −1.24545 −0.05 −9.36811−1.20372 −0.05 −9.42095 −1.13381 −0.05 −9.47609 −1.01007 −0.05 −9.50325−0.82306 −0.05 −9.477 −0.5772 −0.05 −9.3861 −0.28213 −0.05 −9.227620.055253 −0.05 −8.99903 0.429149 −0.05 −8.69796 0.833769 −0.05 −8.321811.2623 −0.05 −7.86774 1.706242 −0.05 −7.33296 2.15483 −0.05 −6.714732.594185 −0.05 −6.01104 3.006511 −0.05 −5.22184 3.369669 −0.05 −4.352513.662426 −0.05 −3.41445 3.871181 −0.05 −2.41776 3.93794 −0.05 −1.388853.845631 −0.05 −0.35937 3.588711 −0.05 0.644165 3.176693 −0.05 1.5975872.621488 −0.05 2.489445 1.949993 −0.05 3.319686 1.192717 −0.05 4.0895750.37256 −0.05 4.803533 −0.49005 −0.05 5.466903 −1.37788 −0.05 6.083446−2.27753 −0.05 6.655406 −3.17828 −0.05 7.188084 −4.0686 −0.05 7.685035−4.93919 −0.05 8.146927 −5.7833 −0.05 8.576585 −6.59355 −0.05 8.976055−7.36372 −0.05 9.345519 −8.08922 −0.05 9.685551 −8.7659 −0.05 9.997578−9.38983 −0.05 10.28085 −9.95892 −0.05 10.53533 −10.4715 −0.05 10.76157−10.9262 −0.05 10.95951 −11.3233 −0.05 11.12947 −11.6632 −0.05 11.27212−11.9475 −0.05 11.38846 −12.1786 −0.05 11.47991 −12.3596 −0.05 11.52386−12.5051 −0.05 11.49556 −12.608 −0.05 11.45438 −12.6606 −0.05 11.41318−12.6907 −0.05 11.37289 −12.7083 −0.05 11.31678 −12.7183 −0.05 11.22792−12.7036 −0.05 11.12586 −12.623 −0.05 11.04504 −12.4671 −0.05 10.94189−12.2683 −0.05 10.81441 −12.0243 −0.05 10.66117 −11.7333 −0.05 10.48099−11.3943 −0.05 10.27283 −11.0073 −0.05 10.03569 −10.5728 −0.05 9.768989−10.0921 −0.05 9.472312 −9.56688 −0.05 9.144215 −9.00035 −0.05 8.784991−8.39525 −0.05 8.393658 −7.75574 −0.05 7.96902 −7.08677 −0.05 7.511449−6.39286 −0.05 7.019665 −5.68025 −0.05 6.492219 −4.956 −0.05 5.929536−4.22641 −0.05 5.330239 −3.49993 −0.05 4.681805 −2.79633 −0.05 3.977849−2.1343 −0.05 3.225818 −1.52309 −0.05 2.434077 −0.97284 −0.05 1.612751−0.49313 −0.05 0.773756 −0.09221 −0.05 −0.07057 0.222513 −0.05 −0.912440.441837 −0.05 −1.73605 0.579694 −0.05 −2.53447 0.638716 −0.05 −3.305410.625575 −0.05 −4.04673 0.54352 −0.05 −4.75051 0.395471 −0.05 −5.406460.192467 −0.05 −6.0072 −0.05053 −0.05 −6.54598 −0.31447 −0.05 −7.02068−0.592 −0.05 −7.45537 −0.83169 −0.05 −7.83884 −1.044 −0.05 −8.1756−1.22144 −0.05 −8.47511 −1.34536 −0.05 −8.73664 −1.40815 −0.05 −8.95248−1.40615 −0.05 −9.11081 −1.36238 −0.05 −9.21803 −1.31547 −0.05 −9.20533−1.01983 0 −9.24971 −0.97839 0 −9.30405 −0.90879 0 −9.36125 −0.78495 0−9.39074 −0.59705 0 −9.3672 −0.34942 0 −9.27969 −0.05139 0 −9.125640.29047 0 −8.90269 0.670794 0 −8.60843 1.084122 0 −8.23998 1.523874 0−7.79409 1.98163 0 −7.26725 2.446483 0 −6.65595 2.904255 0 −5.957263.33663 0 −5.16998 3.720326 0 −4.29777 4.030131 0 −3.35134 4.24553 0−2.34385 4.316407 0 −1.30309 4.221301 0 −0.26311 3.953059 0 0.7464943.52118 0 1.70404 2.946368 0 2.598305 2.255826 0 3.42701 1.477607 04.191935 0.635571 0 4.898232 −0.24911 0 5.551994 −1.15871 0 6.157961−2.07925 0 6.718593 −2.99999 0 7.239622 −3.90936 0 7.725212 −4.79786 08.176288 −5.65866 0 8.595751 −6.48446 0 8.985926 −7.26899 0 9.347037−8.00767 0 9.679889 −8.69623 0 9.985524 −9.33099 0 10.26344 −9.9097 010.51348 −10.4307 0 10.73609 −10.8929 0 10.93111 −11.2963 0 11.09877−11.6415 0 11.23965 −11.9303 0 11.35466 −12.1649 0 11.44512 −12.3487 011.48957 −12.4954 0 11.46068 −12.599 0 11.41875 −12.6519 0 11.3768−12.6819 0 11.3357 −12.6993 0 11.27851 −12.7087 0 11.18853 −12.6914 011.08854 −12.6054 0 11.00778 −12.4472 0 10.90475 −12.2455 0 10.77761−11.9977 0 10.62499 −11.7021 0 10.44586 −11.3576 0 10.23931 −10.964 010.00448 −10.5217 0 9.740927 −10.0319 0 9.448273 −9.49642 0 9.12537−8.91806 0 8.77247 −8.29962 0 8.388751 −7.64512 0 7.972954 −6.95944 07.525378 −6.24712 0 7.044666 −5.51429 0 6.529024 −4.76807 0 5.978714−4.01479 0 5.391985 −3.26293 0 4.759548 −2.52873 0 4.076926 −1.82756 03.348107 −1.16972 0 2.577569 −0.5671 0 1.771346 −0.03193 0 0.9376760.423742 0 0.086014 0.78823 0 −0.77182 1.054605 0 −1.62244 1.219769 0−2.45353 1.284354 0 −3.25437 1.255242 0 −4.01603 1.141302 0 −4.728370.953775 0 −5.38266 0.709915 0 −5.97435 0.42886 0 −6.50105 0.130797 0−6.96391 −0.17247 0 −7.38074 −0.44426 0 −7.74772 −0.68635 0 −8.0703−0.89067 0 −8.35749 −1.0421 0 −8.61045 −1.13304 0 −8.82425 −1.15778 0−8.98662 −1.13154 0 −9.09684 −1.0893 0 −8.9765 −0.52373 0.1 −9.02261−0.48262 0.1 −9.07967 −0.41332 0.1 −9.14064 −0.28885 0.1 −9.17412−0.0986 0.1 −9.155 0.153136 0.1 −9.07299 0.457524 0.1 −8.92612 0.8086130.1 −8.71226 1.201682 0.1 −8.42889 1.631773 0.1 −8.07268 2.09258 0.1−7.63919 2.575657 0.1 −7.12372 3.06983 0.1 −6.52127 3.559962 0.1−5.82717 4.026246 0.1 −5.03848 4.44115 0.1 −4.15658 4.775622 0.1−3.19138 4.997917 0.1 −2.16151 5.068272 0.1 −1.09827 4.958495 0.1−0.0397 4.660737 0.1 0.980285 4.186737 0.1 1.941639 3.567043 0.12.834105 2.830791 0.1 3.654499 2.005929 0.1 4.405529 1.118473 0.15.093647 0.188658 0.1 5.726357 −0.7647 0.1 6.310044 −1.7271 0.1 6.847908−2.68765 0.1 7.345805 −3.63488 0.1 7.808959 −4.5591 0.1 8.238767 −5.45330.1 8.638256 −6.31034 0.1 9.01021 −7.12384 0.1 9.354817 −7.88925 0.19.673305 −8.60217 0.1 9.966118 −9.25918 0.1 10.23323 −9.8578 0.110.47422 −10.3965 0.1 10.68937 −10.8741 0.1 10.87833 −11.2908 0.111.04119 −11.6474 0.1 11.17832 −11.9455 0.1 11.29047 −12.1877 0.111.37879 −12.3774 0.1 11.42431 −12.527 0.1 11.39461 −12.6323 0.111.35111 −12.6857 0.1 11.30756 −12.7156 0.1 11.26473 −12.7323 0.111.20536 −12.7396 0.1 11.11342 −12.7167 0.1 11.01774 −12.6201 0.110.93724 −12.4566 0.1 10.83465 −12.248 0.1 10.70835 −11.9915 0.110.55721 −11.6854 0.1 10.38036 −11.3282 0.1 10.17719 −10.9197 0.19.947108 −10.4601 0.1 9.689942 −9.95044 0.1 9.405368 −9.39242 0.19.092827 −8.7886 0.1 8.752333 −8.14187 0.1 8.383387 −7.45598 0.17.984714 −6.73572 0.1 7.556503 −5.98561 0.1 7.097312 −5.21165 0.16.604732 −4.42103 0.1 6.078717 −3.62012 0.1 5.516888 −2.81744 0.14.914367 −2.02471 0.1 4.268367 −1.25314 0.1 3.57859 −0.51463 0.12.844383 0.180014 0.1 2.064688 0.812969 0.1 1.242155 1.366013 0.10.382156 1.821362 0.1 −0.50134 2.168838 0.1 −1.39555 2.386757 0.1−2.28041 2.470391 0.1 −3.13199 2.425672 0.1 −3.93051 2.269023 0.1−4.66153 2.024029 0.1 −5.31787 1.717577 0.1 −5.89872 1.3755 0.1 −6.407751.020237 0.1 −6.85114 0.670636 0.1 −7.23891 0.344596 0.1 −7.577710.051071 0.1 −7.87474 −0.20168 0.1 −8.13845 −0.40272 0.1 −8.37348−0.54459 0.1 −8.58027 −0.61844 0.1 −8.74807 −0.62551 0.1 −8.86409−0.59237 0.1 −8.76451 0.061449 0.2 −8.81204 0.102729 0.2 −8.871190.172509 0.2 −8.93501 0.298551 0.2 −8.97147 0.492123 0.2 −8.955130.748789 0.2 −8.87639 1.060007 0.2 −8.73372 1.420227 0.2 −8.525081.825102 0.2 −8.24745 2.269744 0.2 −7.89564 2.747766 0.2 −7.466083.250319 0.2 −6.9529 3.764724 0.2 −6.34967 4.275463 0.2 −5.650394.760882 0.2 −4.85137 5.193337 0.2 −3.95386 5.538334 0.2 −2.968095.756637 0.2 −1.91679 5.809273 0.2 −0.83549 5.669362 0.2 0.234285.332283 0.2 1.2583 4.814148 0.2 2.215124 4.144847 0.2 3.096188 3.3583560.2 3.901227 2.485159 0.2 4.63348 1.55011 0.2 5.300178 0.575069 0.25.910287 −0.42239 0.2 6.470667 −1.42677 0.2 6.984495 −2.42678 0.27.459179 −3.41139 0.2 7.90022 −4.37085 0.2 8.309284 −5.29809 0.28.689422 −6.18605 0.2 9.043696 −7.02831 0.2 9.372099 −7.82039 0.29.676228 −8.55774 0.2 9.956162 −9.23711 0.2 10.21236 −9.85583 0.210.4441 −10.4124 0.2 10.6515 −10.9058 0.2 10.83408 −11.3361 0.2 10.99176−11.7042 0.2 11.12478 −12.012 0.2 11.23375 −12.2619 0.2 11.31967−12.4577 0.2 11.36636 −12.6104 0.2 11.3364 −12.7177 0.2 11.29122−12.7717 0.2 11.24593 −12.8013 0.2 11.20118 −12.817 0.2 11.13962 −12.8220.2 11.04629 −12.7927 0.2 10.95521 −12.6863 0.2 10.87521 −12.5167 0.210.77335 −12.3004 0.2 10.64824 −12.0343 0.2 10.49889 −11.7164 0.210.32464 −11.3454 0.2 10.12511 −10.9206 0.2 9.899946 −10.4423 0.29.649241 −9.91121 0.2 9.372736 −9.32918 0.2 9.070415 −8.69841 0.28.741865 −8.02201 0.2 8.386891 −7.30358 0.2 8.004343 −6.54778 0.27.594464 −5.75915 0.2 7.155722 −4.94356 0.2 6.685229 −4.10825 0.26.182678 −3.25961 0.2 5.645357 −2.40586 0.2 5.068803 −1.55763 0.24.451855 −0.72534 0.2 3.790416 0.079065 0.2 3.080835 0.84155 0.22.320055 1.544459 0.2 1.507877 2.168446 0.2 0.648341 2.692995 0.2−0.24955 3.095986 0.2 −1.16915 3.35815 0.2 −2.08561 3.469349 0.2−2.97027 3.43361 0.2 −3.79724 3.269151 0.2 −4.54869 3.004284 0.2−5.21657 2.670169 0.2 −5.80063 2.295981 0.2 −6.30578 1.906597 0.2−6.73939 1.521182 0.2 −7.11009 1.156855 0.2 −7.42868 0.825687 0.2−7.70455 0.537358 0.2 −7.94661 0.298934 0.2 −8.16261 0.118022 0.2−8.35784 0.004845 0.2 −8.52656 −0.03155 0.2 −8.64795 −0.00704 0.2−8.53639 0.696812 0.3 −8.58482 0.738567 0.3 −8.64511 0.809324 0.3−8.7099 0.937395 0.3 −8.74691 1.133857 0.3 −8.73164 1.394441 0.3−8.65463 1.710964 0.3 −8.51423 2.078001 0.3 −8.30797 2.490781 0.3−8.03251 2.945062 0.3 −7.68341 3.434337 0.3 −7.25493 3.949359 0.3−6.74014 4.477114 0.3 −6.13206 5 0.3 −5.42494 5.495464 0.3 −4.614755.93239 0.3 −3.70322 6.274441 0.3 −2.7021 6.480449 0.3 −1.63688 6.5112080.3 −0.54578 6.341731 0.3 0.528494 5.968834 0.3 1.550502 5.411869 0.32.499347 4.703407 0.3 3.367985 3.878637 0.3 4.157879 2.968616 0.34.873078 1.998199 0.3 5.521249 0.989675 0.3 6.111367 −0.03853 0.36.651904 −1.07129 0.3 7.146393 −2.09902 0.3 7.601714 −3.11003 0.38.024091 −4.09433 0.3 8.415544 −5.0448 0.3 8.779185 −5.95448 0.3 9.11832−6.81694 0.3 9.433233 −7.62758 0.3 9.725778 −8.38178 0.3 9.995214−9.07658 0.3 10.24196 −9.70919 0.3 10.4656 −10.2781 0.3 10.66612−10.7824 0.3 10.84294 −11.2221 0.3 10.99589 −11.5983 0.3 11.12511−11.9128 0.3 11.23108 −12.1681 0.3 11.31471 −12.3681 0.3 11.36148−12.5231 0.3 11.33034 −12.6317 0.3 11.28381 −12.6856 0.3 11.2374−12.7146 0.3 11.19137 −12.7295 0.3 11.12842 −12.7325 0.3 11.03434−12.6989 0.3 10.94678 −12.5856 0.3 10.86736 −12.4112 0.3 10.76631−12.1886 0.3 10.64241 −11.9148 0.3 10.49478 −11.5876 0.3 10.32291−11.2054 0.3 10.1266 −10.7676 0.3 9.905676 −10.2744 0.3 9.6604 −9.72650.3 9.390561 −9.12559 0.3 9.096527 −8.4738 0.3 8.778122 −7.77414 0.38.435354 −7.03013 0.3 8.066765 −6.24653 0.3 7.671844 −5.42795 0.37.249762 −4.58016 0.3 6.797284 −3.71042 0.3 6.313839 −2.82529 0.35.79641 −1.93422 0.3 5.240031 −1.04611 0.3 4.641806 −0.17128 0.3 3.99850.678591 0.3 3.304754 1.487397 0.3 2.55663 2.237295 0.3 1.7525062.908154 0.3 0.895221 3.477983 0.3 −0.00737 3.922172 0.3 −0.938374.218845 0.3 −1.87105 4.355477 0.3 −2.77429 4.33508 0.3 −3.619624.175739 0.3 −4.3876 3.906991 0.3 −5.06755 3.561559 0.3 −5.658113.170338 0.3 −6.16374 2.759753 0.3 −6.59296 2.351227 0.3 −6.955511.961769 0.3 −7.26109 1.603903 0.3 −7.52098 1.288402 0.3 −7.745021.023225 0.3 −7.94319 0.815539 0.3 −8.12458 0.674077 0.3 −8.290260.613025 0.3 −8.41604 0.628604 0.3 −8.31163 1.368225 0.4 −8.360241.410573 0.4 −8.42039 1.482439 0.4 −8.48386 1.612612 0.4 −8.518961.811376 0.4 −8.503 2.074574 0.4 −8.4265 2.39449 0.4 −8.28699 2.7656350.4 −8.08129 3.182455 0.4 −7.80537 3.640689 0.4 −7.45436 4.133033 0.4−7.02237 4.649315 0.4 −6.50219 5.180302 0.4 −5.88667 5.70482 0.4−5.16994 6.198895 0.4 −4.3487 6.629152 0.4 −3.4259 6.958665 0.4 −2.415227.145692 0.4 −1.34452 7.151097 0.4 −0.25347 6.951945 0.4 0.8178916.547605 0.4 1.833374 5.959649 0.4 2.772134 5.221757 0.4 3.6281364.369376 0.4 4.403871 3.433185 0.4 5.103769 2.437754 0.4 5.7358981.407341 0.4 6.309574 0.36226 0.4 6.835165 −0.68701 0.4 7.315917−1.73403 0.4 7.757615 −2.76572 0.4 8.166746 −3.76697 0.4 8.545517−4.73324 0.4 8.897081 −5.65769 0.4 9.224894 −6.53395 0.4 9.529323−7.35737 0.4 9.812212 −8.12339 0.4 10.07279 −8.82902 0.4 10.31091−9.47141 0.4 10.52707 −10.0491 0.4 10.72111 −10.5611 0.4 10.89243−11.0076 0.4 11.04079 −11.3895 0.4 11.16626 −11.7087 0.4 11.26925−11.968 0.4 11.35058 −12.1711 0.4 11.39682 −12.3278 0.4 11.36435 −12.4370.4 11.31657 −12.4905 0.4 11.26923 −12.5188 0.4 11.22213 −12.5328 0.411.15807 −12.5343 0.4 11.06352 −12.4972 0.4 10.97904 −12.3788 0.410.90042 −12.2004 0.4 10.80043 −11.9728 0.4 10.67799 −11.6927 0.410.53232 −11.3578 0.4 10.36301 −10.9667 0.4 10.16999 −10.5185 0.49.953245 −10.0134 0.4 9.713159 −9.45199 0.4 9.449556 −8.83607 0.49.163102 −8.16763 0.4 8.853536 −7.44971 0.4 8.52103 −6.68576 0.48.164029 −5.88052 0.4 7.780482 −5.03859 0.4 7.370877 −4.16545 0.46.931682 −3.26641 0.4 6.462046 −2.35223 0.4 5.958652 −1.43518 0.45.416428 −0.52056 0.4 4.832063 0.381528 0.4 4.203616 1.265706 0.43.523424 2.10962 0.4 2.78661 2.895038 0.4 1.990469 3.601572 0.4 1.1369684.206367 0.4 0.233247 4.683851 0.4 −0.70288 5.009871 0.4 −1.64435.169944 0.4 −2.5597 5.165921 0.4 −3.41888 5.015559 0.4 −4.20087 4.748940.4 −4.89205 4.399598 0.4 −5.48972 3.999583 0.4 −5.99779 3.576491 0.4−6.42557 3.152834 0.4 −6.78348 2.746779 0.4 −7.0818 2.371529 0.4−7.33228 2.038266 0.4 −7.54489 1.758445 0.4 −7.72952 1.53486 0.4−7.89898 1.374388 0.4 −8.05993 1.293994 0.4 −8.1885 1.300534 0.4−8.09223 2.049641 0.5 −8.14028 2.092958 0.5 −8.19911 2.16638 0.5−8.26029 2.298128 0.5 −8.29402 2.497489 0.5 −8.27773 2.76075 0.5−8.20196 3.080921 0.5 −8.06359 3.452959 0.5 −7.85871 3.872199 0.5−7.58259 4.332936 0.5 −7.22976 4.827793 0.5 −6.79398 5.346827 0.5−6.26781 5.872199 0.5 −5.64422 6.386734 0.5 −4.91759 6.865685 0.5−4.08555 7.277528 0.5 −3.15226 7.584111 0.5 −2.13265 7.74441 0.5−1.05627 7.720705 0.5 0.03624 7.492463 0.5 1.100733 7.06158 0.5 2.1050166.450807 0.5 3.031326 5.693408 0.5 3.874484 4.824252 0.5 4.6372953.87409 0.5 5.324169 2.866863 0.5 5.943351 1.823325 0.5 6.5043750.760513 0.5 7.017028 −0.30328 0.5 7.483445 −1.35838 0.5 7.911366−2.39409 0.5 8.307441 −3.40309 0.5 8.673931 −4.37685 0.5 9.013901−5.30844 0.5 9.330835 −6.19148 0.5 9.625103 −7.02129 0.5 9.898545−7.79329 0.5 10.15049 −8.50448 0.5 10.38188 −9.15198 0.5 10.59192−9.73426 0.5 10.78043 −10.2504 0.5 10.94685 −10.7006 0.5 11.09097−11.0858 0.5 11.21284 −11.4077 0.5 11.31286 −11.6691 0.5 11.39185−11.8739 0.5 11.43601 −12.0318 0.5 11.40178 −12.1408 0.5 11.35308−12.1937 0.5 11.30524 −12.2211 0.5 11.25753 −12.2345 0.5 11.19296−12.2345 0.5 11.09841 −12.1949 0.5 11.01597 −12.0729 0.5 10.93884−11.8916 0.5 10.84074 −11.6603 0.5 10.72064 −11.3756 0.5 10.57783−11.0352 0.5 10.41188 −10.6377 0.5 10.22279 −10.1821 0.5 10.01054−9.66856 0.5 9.775532 −9.09782 0.5 9.517623 −8.47153 0.5 9.237513−7.79174 0.5 8.934858 −7.06156 0.5 8.609929 −6.28447 0.5 8.261292−5.46521 0.5 7.888912 −4.60852 0.5 7.49132 −3.72036 0.5 7.064874−2.80913 0.5 6.608597 −1.87938 0.5 6.119091 −0.93922 0.5 5.5912620.001346 0.5 5.021267 0.93162 0.5 4.404375 1.836321 0.5 3.7354192.702838 0.5 3.008624 3.512704 0.5 2.220445 4.244505 0.5 1.3722144.874943 0.5 0.470846 5.378122 0.5 −0.46898 5.728556 0.5 −1.419485.910297 0.5 −2.34657 5.923921 0.5 −3.21824 5.786394 0.5 −4.010465.527659 0.5 −4.71092 5.181564 0.5 −5.31635 4.780831 0.5 −5.829964.353863 0.5 −6.25899 3.924009 0.5 −6.61354 3.510208 0.5 −6.905253.127076 0.5 −7.14617 2.785475 0.5 −7.3481 2.490866 0.5 −7.523492.252611 0.5 −7.68302 2.07857 0.5 −7.83752 1.985693 0.5 −7.967281.983004 0.5 −7.91603 2.737665 0.6 −7.96273 2.782164 0.6 −8.018732.857437 0.6 −8.07565 2.990506 0.6 −8.10548 3.189593 0.6 −8.087183.451455 0.6 −8.01089 3.769741 0.6 −7.87214 4.139344 0.6 −7.665944.555468 0.6 −7.38668 5.01122 0.6 −7.02832 5.498272 0.6 −6.58441 6.005720.6 −6.04808 6.51524 0.6 −5.41321 7.009134 0.6 −4.67554 7.462517 0.6−3.83446 7.844391 0.6 −2.89599 8.117977 0.6 −1.87617 8.244565 0.6−0.80538 8.18942 0.6 0.277412 7.93471 0.6 1.329298 7.483372 0.6 2.3201676.858016 0.6 3.233631 6.090819 0.6 4.064986 5.215302 0.6 4.8169694.261058 0.6 5.493938 3.251397 0.6 6.104031 2.206333 0.6 6.656661.142053 0.6 7.160591 0.077935 0.6 7.618378 −0.97495 0.6 8.038129−2.00825 0.6 8.426311 −3.01646 0.6 8.785227 −3.99003 0.6 9.117893−4.92191 0.6 9.427755 −5.8056 0.6 9.715298 −6.6363 0.6 9.982379 −7.409330.6 10.22828 −8.12169 0.6 10.4544 −8.77046 0.6 10.65939 −9.35405 0.610.84304 −9.87154 0.6 11.005 −10.323 0.6 11.14511 −10.7093 0.6 11.26347−11.0322 0.6 11.36055 −11.2946 0.6 11.43717 −11.5001 0.6 11.4781−11.6585 0.6 11.44179 −11.7669 0.6 11.39241 −11.8189 0.6 11.34438−11.8456 0.6 11.29642 −11.8582 0.6 11.23176 −11.8571 0.6 11.13757−11.8154 0.6 11.05643 −11.6911 0.6 10.98128 −11.5078 0.6 10.88565−11.2739 0.6 10.76853 −10.986 0.6 10.62915 −10.642 0.6 10.46711 −10.24010.6 10.28234 −9.77971 0.6 10.07479 −9.26083 0.6 9.84478 −8.68427 0.69.592194 −8.05171 0.6 9.317569 −7.36531 0.6 9.020525 −6.62826 0.68.701274 −5.84412 0.6 8.358382 −5.01777 0.6 7.992818 −4.15418 0.67.602506 −3.2594 0.6 7.18393 −2.34259 0.6 6.735794 −1.40731 0.6 6.254997−0.45865 0.6 5.736202 0.491949 0.6 5.175157 1.433307 0.6 4.5668692.348663 0.6 3.906474 3.226822 0.6 3.188069 4.049463 0.6 2.4079744.795489 0.6 1.567059 5.441694 0.6 0.671704 5.96225 0.6 −0.264116.331963 0.6 −1.2132 6.534386 0.6 −2.14173 6.568958 0.6 −3.017996.451026 0.6 −3.81607 6.209218 0.6 −4.52433 5.87686 0.6 −5.138315.486531 0.6 −5.66032 5.066876 0.6 −6.09625 4.641732 0.6 −6.455094.230101 0.6 −6.74844 3.846985 0.6 −6.98871 3.503969 0.6 −7.187883.206888 0.6 −7.35849 2.964855 0.6 −7.51207 2.785815 0.6 −7.660992.684396 0.6 −7.79035 2.672343 0.6 −7.77011 3.424426 0.7 −7.814723.470111 0.7 −7.86661 3.547221 0.7 −7.91749 3.681575 0.7 −7.941623.880313 0.7 −7.92011 4.140697 0.7 −7.84244 4.457057 0.7 −7.702274.823942 0.7 −7.49322 5.234899 0.7 −7.20872 5.682364 0.7 −6.84226.156413 0.7 −6.38724 6.644421 0.7 −5.83805 7.130508 0.7 −5.189957.594775 0.7 −4.44056 8.012658 0.7 −3.59148 8.354456 0.7 −2.651058.586024 0.7 −1.63683 8.672219 0.7 −0.57966 8.582838 0.7 0.4845928.302922 0.7 1.517982 7.835809 0.7 2.492086 7.202766 0.7 3.391166.433867 0.7 4.21075 5.56058 0.7 4.953115 4.61081 0.7 5.622457 3.6070230.7 6.226522 2.569359 0.7 6.774253 1.515011 0.7 7.273499 0.457971 0.77.728046 −0.59164 0.7 8.144741 −1.62239 0.7 8.529593 −2.62476 0.78.884855 −3.59194 0.7 9.21356 −4.51721 0.7 9.519073 −5.39436 0.79.802094 −6.21871 0.7 10.06456 −6.98576 0.7 10.30576 −7.69256 0.710.52628 −8.33636 0.7 10.72601 −8.91553 0.7 10.9047 −9.42919 0.711.06213 −9.87736 0.7 11.19818 −10.2609 0.7 11.31304 −10.5816 0.711.40717 −10.8423 0.7 11.48142 −11.0464 0.7 11.5193 −11.2042 0.711.48097 −11.3104 0.7 11.43086 −11.3609 0.7 11.38259 −11.3866 0.711.33432 −11.3986 0.7 11.26949 −11.3968 0.7 11.17562 −11.3547 0.711.09589 −11.2307 0.7 11.02263 −11.0473 0.7 10.92939 −10.8133 0.710.81516 −10.5255 0.7 10.67916 −10.1815 0.7 10.52097 −9.77969 0.710.34048 −9.31944 0.7 10.13762 −8.80081 0.7 9.912606 −8.22459 0.79.665372 −7.5924 0.7 9.39632 −6.9064 0.7 9.105053 −6.1697 0.7 8.791783−5.38575 0.7 8.455032 −4.55925 0.7 8.093947 −3.69501 0.7 7.708037−2.79845 0.7 7.293935 −1.87638 0.7 6.849884 −0.93588 0.7 6.3733390.014223 0.7 5.858844 0.964239 0.7 5.301955 1.903477 0.7 4.6995972.820776 0.7 4.045806 3.70133 0.7 3.334958 4.527488 0.7 2.5636385.278966 0.7 1.732393 5.933165 0.7 0.846946 6.4649 0.7 −0.07821 6.850410.7 −1.01851 7.073401 0.7 −1.9423 7.132093 0.7 −2.81912 7.039743 0.7−3.62322 6.822719 0.7 −4.33939 6.512906 0.7 −4.96195 6.142018 0.7−5.4922 5.738441 0.7 −5.93662 5.32617 0.7 −6.30388 4.924131 0.7 −6.604254.547478 0.7 −6.85004 4.208433 0.7 −7.05251 3.915234 0.7 −7.222753.674035 0.7 −7.37359 3.491776 0.7 −7.51716 3.381877 0.7 −7.644663.360453 0.7 −7.69166 4.090324 0.8 −7.73341 4.137461 0.8 −7.779824.216486 0.8 −7.82254 4.351428 0.8 −7.83951 4.54778 0.8 −7.813874.803642 0.8 −7.73406 5.11412 0.8 −7.59144 5.473327 0.8 −7.37796 5.87390.8 −7.08597 6.306576 0.8 −6.70866 6.759994 0.8 −6.24027 7.220434 0.8−5.67637 7.671441 0.8 −5.01441 8.093343 0.8 −4.2546 8.462907 0.8−3.40123 8.753034 0.8 −2.46448 8.933703 0.8 −1.4631 8.974528 0.8−0.42555 8.850705 0.8 0.615777 8.549603 0.8 1.627056 8.074655 0.82.582612 7.4443 0.8 3.467654 6.685227 0.8 4.277434 5.826056 0.8 5.0131344.892498 0.8 5.67851 3.906118 0.8 6.280656 2.886345 0.8 6.8277851.849863 0.8 7.327084 0.810309 0.8 7.782093 −0.22201 0.8 8.199303−1.23651 0.8 8.5844 −2.22261 0.8 8.939468 −3.17473 0.8 9.267532 −4.08610.8 9.57179 −4.95054 0.8 9.853027 −5.76332 0.8 10.11256 −6.52017 0.810.35065 −7.2178 0.8 10.56775 −7.85355 0.8 10.76384 −8.4257 0.8 10.93872−8.93337 0.8 11.09241 −9.37646 0.8 11.22492 −9.75581 0.8 11.33655−10.0731 0.8 11.42788 −10.331 0.8 11.49983 −10.5331 0.8 11.53302−10.6897 0.8 11.49204 −10.7934 0.8 11.44156 −10.8423 0.8 11.39356−10.8669 0.8 11.34559 −10.8782 0.8 11.28135 −10.876 0.8 11.18833 −10.8340.8 11.10869 −10.7112 0.8 11.03796 −10.5285 0.8 10.94785 −10.2954 0.810.83728 −10.0087 0.8 10.70542 −9.66612 0.8 10.55173 −9.26621 0.810.37598 −8.80828 0.8 10.17795 −8.29249 0.8 9.957681 −7.7197 0.89.715111 −7.09154 0.8 9.450263 −6.41033 0.8 9.162766 −5.67915 0.88.852707 −4.90154 0.8 8.518652 −4.08218 0.8 8.159926 −3.226 0.8 7.775533−2.33849 0.8 7.362435 −1.42541 0.8 6.918673 −0.49509 0.8 6.4422980.444823 0.8 5.928413 1.384484 0.8 5.372403 2.313063 0.8 4.7716743.219677 0.8 4.121142 4.090194 0.8 3.415765 4.907674 0.8 2.652525.652736 0.8 1.832052 6.303912 0.8 0.960098 6.837969 0.8 0.0501217.232978 0.8 −0.87557 7.473551 0.8 −1.78747 7.55692 0.8 −2.657587.494564 0.8 −3.46043 7.309983 0.8 −4.18093 7.032479 0.8 −4.812316.69197 0.8 −5.35448 6.315795 0.8 −5.81212 5.927418 0.8 −6.193065.546175 0.8 −6.50696 5.187389 0.8 −6.76435 4.862819 0.8 −6.976214.581291 0.8 −7.15238 4.348663 0.8 −7.30442 4.171358 0.8 −7.443424.059128 0.8 −7.56747 4.028308 0.8 −7.69295 4.656167 0.9 −7.731144.704676 0.9 −7.77094 4.785235 0.9 −7.80418 4.919972 0.9 −7.812955.11304 0.9 −7.78251 5.363415 0.9 −7.69998 5.666868 0.9 −7.554146.016786 0.9 −7.33509 6.404294 0.9 −7.03426 6.81825 0.9 −6.644877.245549 0.9 −6.16227 7.671274 0.9 −5.58397 8.078619 0.9 −4.909928.448704 0.9 −4.14317 8.760509 0.9 −3.29056 8.990668 0.9 −2.363679.114491 0.9 −1.38157 9.107446 0.9 −0.36897 8.950498 0.9 0.645528.633112 0.9 1.632862 8.158039 0.9 2.570019 7.539793 0.9 3.4427916.800772 0.9 4.245725 5.966172 0.9 4.978477 5.059115 0.9 5.6441674.100216 0.9 6.249043 3.108182 0.9 6.800386 2.099144 0.9 7.3045571.086435 0.9 7.764782 0.08033 0.9 8.187122 −0.90857 0.9 8.576802−1.87778 0.9 8.935715 −2.81331 0.9 9.266768 −3.70863 0.9 9.572941−4.55778 0.9 9.855158 −5.35616 0.9 10.11408 −6.09979 0.9 10.35104−6.78521 0.9 10.5661 −7.41002 0.9 10.75963 −7.97243 0.9 10.93148−8.47162 0.9 11.082 −8.90741 0.9 11.21134 −9.28062 0.9 11.31998 −9.592860.9 11.40865 −9.8467 0.9 11.47836 −10.0456 0.9 11.50584 −10.2005 0.911.46184 −10.3008 0.9 11.41119 −10.3475 0.9 11.36373 −10.371 0.911.31637 −10.3818 0.9 11.25308 −10.3794 0.9 11.16113 −10.3386 0.911.08085 −10.2186 0.9 11.01297 −10.0375 0.9 10.92639 −9.80659 0.910.81989 −9.52266 0.9 10.69254 −9.18346 0.9 10.54366 −8.78761 0.910.3728 −8.33445 0.9 10.17956 −7.8242 0.9 9.963696 −7.25775 0.9 9.725163−6.63664 0.9 9.463446 −5.96327 0.9 9.178226 −5.24061 0.9 8.86941−4.47211 0.9 8.535612 −3.66229 0.9 8.175973 −2.816 0.9 7.789487 −1.939080.9 7.373471 −1.04467 0.9 6.925737 −0.13375 0.9 6.445019 0.786165 0.95.927176 1.70544 0.9 5.367523 2.613056 0.9 4.7641 3.498439 0.9 4.1129894.348247 0.9 3.410043 5.146417 0.9 2.652924 5.874713 0.9 1.8426396.513133 0.9 0.985206 7.040967 0.9 0.093161 7.438834 0.9 −0.813627.692891 0.9 −1.70865 7.799824 0.9 −2.5666 7.769307 0.9 −3.364027.621539 0.9 −4.08549 7.382796 0.9 −4.72359 7.080592 0.9 −5.276956.740657 0.9 −5.74881 6.385438 0.9 −6.14579 6.033921 0.9 −6.476245.701231 0.9 −6.74894 5.398615 0.9 −6.97413 5.135428 0.9 −7.160144.917099 0.9 −7.31661 4.749046 0.9 −7.45177 4.636549 0.9 −7.570934.596458 0.9 −7.75919 5.130638 1 −7.79333 5.180494 1 −7.82572 5.262319 1−7.84858 5.396295 1 −7.84846 5.585546 1 −7.81276 5.829869 1 −7.727176.125539 1 −7.57758 6.464987 1 −7.35227 6.837578 1 −7.04178 7.230005 1−6.63975 7.627189 1 −6.14297 8.012946 1 −5.55133 8.370353 1 −4.867618.681801 1 −4.09767 8.929194 1 −3.25061 9.093796 1 −2.33884 9.157089 1−1.38084 9.101147 1 −0.39654 8.912667 1 0.589206 8.582949 1 1.5523698.113279 1 2.472271 7.513451 1 3.334927 6.80095 1 4.13384 5.997182 14.866832 5.122412 1 5.53624 4.196267 1 6.14735 3.236686 1 6.7064132.259147 1 7.218744 1.276406 1 7.687344 0.298595 1 8.117854 −0.66392 18.514898 −1.60235 1 8.880243 −2.50973 1 9.216521 −3.37942 1 9.526494−4.20538 1 9.811312 −4.98283 1 10.07123 −5.70792 1 10.30839 −6.37677 110.52244 −6.98709 1 10.71416 −7.53689 1 10.88355 −8.02529 1 11.03129−8.45195 1 11.15773 −8.81756 1 11.26356 −9.12361 1 11.34967 −9.37252 111.41722 −9.56767 1 11.43831 −9.72033 1 11.39105 −9.81658 1 11.34035−9.8611 1 11.29359 −9.88339 1 11.247 −9.89369 1 11.18482 −9.89148 111.09402 −9.8524 1 11.01272 −9.73604 1 10.94789 −9.55731 1 10.86505−9.32942 1 10.76284 −9.04936 1 10.64018 −8.715 1 10.49623 −8.32504 110.33031 −7.87898 1 10.14176 −7.37716 1 9.930007 −6.82064 1 9.695022−6.21101 1 9.43565 −5.55097 1 9.151617 −4.84355 1 8.842577 −4.09239 18.507147 −3.30212 1 8.144237 −2.47785 1 7.75331 −1.62494 1 7.331949−0.75025 1 6.877628 0.138874 1 6.389753 1.035354 1 5.864948 1.929964 15.298555 2.811642 1 4.689354 3.670271 1 4.034713 4.493368 1 3.3315825.265892 1 2.578556 5.970951 1 1.777199 6.59021 1 0.934028 7.105821 10.060873 7.50151 1 −0.82468 7.765511 1 −1.69984 7.894473 1 −2.541697.896344 1 −3.32983 7.787844 1 −4.04878 7.592035 1 −4.69097 7.333602 1−5.25407 7.036234 1 −5.74003 6.720978 1 −6.15379 6.405697 1 −6.502146.104998 1 −6.79224 5.829668 1 −7.03324 5.589498 1 −7.23171 5.389383 1−7.39489 5.233582 1 −7.52682 5.122243 1 −7.64003 5.073235 1 −7.803085.359691 1.05 −7.83507 5.410254 1.05 −7.86355 5.492711 1.05 −7.880955.626214 1.05 −7.87617 5.813342 1.05 −7.83771 6.054327 1.05 −7.750486.345695 1.05 −7.59891 6.679357 1.05 −7.37031 7.043734 1.05 −7.05487.424372 1.05 −6.64622 7.805155 1.05 −6.14221 8.169252 1.05 −5.54398.499697 1.05 −4.85559 8.779586 1.05 −4.08466 8.99242 1.05 −3.241389.121972 1.05 −2.33855 9.153055 1.05 −1.39424 9.071284 1.05 −0.425688.866389 1.05 0.544299 8.530587 1.05 1.494396 8.064375 1.05 2.4051417.474991 1.05 3.262609 6.777287 1.05 4.059702 5.990644 1.05 4.7932325.133807 1.05 5.465096 4.225887 1.05 6.080054 3.28443 1.05 6.6437842.324573 1.05 7.161067 1.35882 1.05 7.634742 0.397245 1.05 8.070214−0.54984 1.05 8.471781 −1.47366 1.05 8.841134 −2.36734 1.05 9.180749−3.2243 1.05 9.493261 −4.03856 1.05 9.779941 −4.80528 1.05 10.04081−5.52072 1.05 10.27845 −6.18084 1.05 10.49228 −6.78344 1.05 10.68334−7.32649 1.05 10.85167 −7.80905 1.05 10.99813 −8.23074 1.05 11.1232−8.59218 1.05 11.22766 −8.89483 1.05 11.31252 −9.14102 1.05 11.37899−9.33408 1.05 11.39665 −9.48548 1.05 11.34768 −9.57954 1.05 11.29701−9.62287 1.05 11.25067 −9.64456 1.05 11.20455 −9.65464 1.05 11.14303−9.65256 1.05 11.05287 −9.61454 1.05 10.9709 −9.50032 1.05 10.90767−9.32296 1.05 10.82681 −9.09683 1.05 10.72684 −8.81902 1.05 10.60663−8.48742 1.05 10.46525 −8.10079 1.05 10.30185 −7.65872 1.05 10.11565−7.16156 1.05 9.905917 −6.61046 1.05 9.672597 −6.00699 1.05 9.414193−5.354 1.05 9.130458 −4.65447 1.05 8.820908 −3.9121 1.05 8.484163−3.13152 1.05 8.119024 −2.31788 1.05 7.725199 −1.47643 1.05 7.30042−0.61391 1.05 6.842 0.26212 1.05 6.349703 1.144767 1.05 5.8205992.024986 1.05 5.250072 2.891674 1.05 4.637299 3.734938 1.05 3.9803734.542765 1.05 3.276858 5.300665 1.05 2.525872 5.992468 1.05 1.7293136.600755 1.05 0.894002 7.10924 1.05 0.031403 7.503325 1.05 −0.842157.772365 1.05 −1.70588 7.913004 1.05 −2.53823 7.932284 1.05 −3.320617.845051 1.05 −4.03756 7.672612 1.05 −4.68147 7.438097 1.05 −5.249487.164064 1.05 −5.74283 6.870773 1.05 −6.16554 6.575496 1.05 −6.523576.292559 1.05 −6.82314 6.032468 1.05 −7.07279 5.805256 1.05 −7.278185.61554 1.05 −7.44522 5.466949 1.05 −7.57559 5.356875 1.05 −7.685555.303462 1.05 −7.84698 5.588744 1.1 −7.87681 5.640015 1.1 −7.901385.723102 1.1 −7.91332 5.856133 1.1 −7.90387 6.041137 1.1 −7.862656.278785 1.1 −7.7738 6.565851 1.1 −7.62025 6.893727 1.1 −7.388367.249891 1.1 −7.06782 7.618739 1.1 −6.6527 7.98312 1.1 −6.14145 8.325561.1 −5.53647 8.629042 1.1 −4.84357 8.87737 1.1 −4.07165 9.055644 1.1−3.23215 9.150147 1.1 −2.33826 9.149022 1.1 −1.40765 9.041422 1.1−0.45482 8.82011 1.1 0.499391 8.478226 1.1 1.436417 8.01546 1.1 2.3379917.436504 1.1 3.190252 6.753574 1.1 3.985503 5.984036 1.1 4.7195475.145115 1.1 5.393848 4.255412 1.1 6.012635 3.332076 1.1 6.5810182.389908 1.1 7.103245 1.441159 1.1 7.581989 0.495846 1.1 8.022422−0.43577 1.1 8.428512 −1.34501 1.1 8.801875 −2.22504 1.1 9.14483−3.06931 1.1 9.459889 −3.87188 1.1 9.748435 −4.62789 1.1 10.01026−5.33368 1.1 10.24838 −5.98507 1.1 10.46201 −6.57997 1.1 10.6524−7.11625 1.1 10.81966 −7.59297 1.1 10.96485 −8.00968 1.1 11.08854−8.36696 1.1 11.19164 −8.66619 1.1 11.27525 −8.90967 1.1 11.34065−9.10062 1.1 11.35487 −9.25076 1.1 11.30418 −9.34264 1.1 11.25355−9.38476 1.1 11.20763 −9.40586 1.1 11.16198 −9.41571 1.1 11.10112−9.41376 1.1 11.01162 −9.37681 1.1 10.92896 −9.26473 1.1 10.86734−9.08874 1.1 10.78845 −8.86439 1.1 10.69074 −8.58882 1.1 10.57299−8.25999 1.1 10.43416 −7.87671 1.1 10.27329 −7.43861 1.1 10.08946−6.94612 1.1 9.881737 −6.40043 1.1 9.650084 −5.80313 1.1 9.392646−5.15717 1.1 9.109207 −4.46552 1.1 8.799144 −3.73192 1.1 8.461085 −2.9611.1 8.093714 −2.15795 1.1 7.69699 −1.32793 1.1 7.268792 −0.47763 1.16.806272 0.385293 1.1 6.309557 1.25409 1.1 5.776163 2.119911 1.1 5.201512.971609 1.1 4.585179 3.799518 1.1 3.925982 4.592088 1.1 3.2220995.335383 1.1 2.473169 6.01395 1.1 1.681418 6.611283 1.1 0.8539757.112656 1.1 0.001932 7.505141 1.1 −0.85961 7.779219 1.1 −1.711917.931535 1.1 −2.53477 7.968223 1.1 −3.31139 7.902259 1.1 −4.026357.753189 1.1 −4.67198 7.542593 1.1 −5.2449 7.291893 1.1 −5.745647.020568 1.1 −6.1773 6.745294 1.1 −6.545 6.48012 1.1 −6.85403 6.2352691.1 −7.11235 6.021015 1.1 −7.32466 5.841696 1.1 −7.49555 5.700315 1.1−7.62436 5.591506 1.1 −7.73107 5.533689 1.1 −7.80758 5.55128 1.1

In the above table, the plane Zadim=0 corresponds to the reference planeP0 situated at the base of the profile. This plane P0 is theintersection of the stacking axis AE of the set of blades with theaxisymmetric surface of the hub, i.e. it passes via the intersectionbetween the axis AE and the surface 12A where it is assembled with theplatform 12. The plane Zadim=1 corresponds to the reference plane P1situated at the top of the profile. This plane P1 is the intersectionbetween the axis AE and the axisymmetric surface of the casing, i.e. itpasses via the intersection of the axis AE with the surface 16A where itis assembled with the top platform 16.

By convention, the stacking axis AE of the set of blades, for the bladeof a rotary wheel, is the axis that extends in the radial direction Zand passes through the barycenter of the profile of the blade in sectionin the X, Y planes. This is the axis on which the various X,Y sectionsof the set of blades are “stacked” when designing the set of blades.

As mentioned at the beginning of the present description, theaerodynamic profile of the invention is substantially identical to thenominal profile defined in the above table, i.e. it departs from saidnominal profile by very little at most, being defined in particularwithin an envelope of ±1 mm in a direction normal to the surface of thenominal profile, and/or having X,Y coordinates lying within a range of±5% relative to the coordinates X,Y of the nominal profile.

The coordinates of the above table are given starting from a valueZadim=0 in the reference plane P0. The coordinate Zadim isnon-dimensional, i.e. for a point P, situated at a distance D from theplane P0 (where D is measured along the axis Z), the value of Zadim isD/H, where H represents the total height of the profile as measuredbetween the planes P0 and P1. Naturally, in the table above, bymultiplying the coordinate Zadim by the height H, it is possible toobtain the complete coordinates of the blade.

Preferably, the height H of the profile of the blade lies in the range120 mm±10%. By way of example, the reference plane P0 is at a distanceof about 400 mm±10% from the axis A of the turbine.

In the above table, the profile is characterized by fifteen sectionplanes at constant Zadim coordinates, for which the coordinates X and Yare specified. These fifteen section planes include ten section planeslocated between Zadim=0 and Zadim=1, and thus arranged in the fluidstream, these section planes being distributed every 10% on the vane'sheight from P0 to P1. The fifteen section planes also include fivesection planes which are substantially outside the fluid stream (threesection planes with Zadim<0, and two section planes with Zadim>1), thesefive section planes being provided for ensuring geometric continuity ofthe stacking, close to the head and the foot, respectively. In eachsection plane at constant coordinate Zadim, the section of the profileis given by a continuous and smooth curve, interconnecting all of thepoints (X,Y). In each section plane, the profile is interpolated so asto generate a uniform profile.

It is advantageous for the rotary wheel that includes the blade of theinvention to have 130 to 150 blades that present aerodynamic profiles asdefined above.

We claim:
 1. An aerodynamic profile for a turbine blade, the profilebeing, when cold and in a non-coated state, substantially identical to anominal profile determined by the Cartesian coordinates X,Y,Zadim givenin Table 1, in which the coordinate Zadim is the quotient D/H, where Dis the distance of the point under consideration from a first referenceX,Y plane situated at the base of the nominal profile, and H is theheight of said profile measured from said first reference plane that isthe intersection of the stacking axis of the set of blades and theaxisymmetric surface of the hub, out to a second reference plane that isthe intersection of said stacking axis with the axisymmetric surface ofthe casing, the measurements D and H being taken radially relative tothe axis of the turbine, while the coordinate X is measured in the axialdirection of the turbine.
 2. The aerodynamic profile as claimed in claim1, wherein said profile is defined within an envelope of ±1 mm in adirection normal to the surface of the nominal profile.
 3. Theaerodynamic profile as claimed in claim 1, wherein the coordinates X,Yof said profile lie within a range of ±5% relative to the coordinatesX,Y of the nominal profile.
 4. The aerodynamic profile as claimed inclaim 1, wherein the blade is a blade of a rotary wheel forming aportion of a rotor of a turbine.
 5. The aerodynamic profile as claimedin claim 4, wherein the blade is a blade of the rotary wheel of thefourth stage of the turbine.
 6. The aerodynamic profile as claimed inclaim 4, wherein the blade is a blade of the rotary wheel of the fourthstage of a turbine having seven stages.
 7. A turbine blade, presentingan aerodynamic profile as claimed in claim
 1. 8. A turbine, includingturbine blades presenting aerodynamic profiles as claimed in claim
 1. 9.A turbine as claimed in claim 8, including a rotary wheel, having 130 to150 blades that present aerodynamic profiles comprising: when cold andin a non-coated state, substantially identical to a nominal profiledetermined by the Cartesian coordinates X,Y,Zadim given in Table 1, inwhich the coordinate Zadim is the quotient D/H, where D is the distanceof the point under consideration from a first reference X,Y planesituated at the base of the nominal profile, and H is the height of saidprofile measured from said first reference plane that is theintersection of the stacking axis of the set of blades and theaxisymmetric surface of the hub, out to a second reference plane that isthe intersection of said stacking axis with the axisymmetric surface ofthe casing, the measurements D and H being taken radially relative tothe axis of the turbine, while the coordinate X is measured in the axialdirection of the turbine.
 10. A rotary wheel of a turbine forming aportion of a turbine rotor, wherein all the blades of this rotary wheelpresent an aerodynamic profile as claimed in claim
 1. 11. The rotarywheel of a turbine as claimed in claim 10, including 130 to 150 bladesthat present aerodynamic profiles comprising: when cold and in anon-coated state, substantially identical to a nominal profiledetermined by the Cartesian coordinates X,Y,Zadim given in Table 1, inwhich the coordinate Zadim is the quotient D/H, where D is the distanceof the point under consideration from a first reference X,Y planesituated at the base of the nominal profile, and H is the height of saidprofile measured from said first reference plane that is theintersection of the stacking axis of the set of blades and theaxisymmetric surface of the hub, out to a second reference plane that isthe intersection of said stacking axis with the axisymmetric surface ofthe casing, the measurements D and H being taken radially relative tothe axis of the turbine, while the coordinate X is measured in the axialdirection of the turbine.